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Publication Detail
AGAINST CUMULATIVE TYPE THEORY
  • Publication Type:
    Journal article
  • Authors:
    BUTTON TIM, TRUEMAN R
  • Publisher:
    Cambridge University Press (CUP)
  • Publication date:
    12/2022
  • Pagination:
    907, 949
  • Journal:
    The Review of Symbolic Logic
  • Volume:
    15
  • Issue:
    4
  • Status:
    Published
  • Print ISSN:
    1755-0203
  • Language:
    en
Abstract
AbstractStandard Type Theory, ${\textrm {STT}}$ , tells us that $b^n(a^m)$ is well-formed iff $n=m+1$ . However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, $\textrm {CTT}$ , which has more relaxed type-restrictions: according to $\textrm {CTT}$ , $b^\beta (a^\alpha )$ is well-formed iff $\beta>\alpha $ . In this paper, we set ourselves against $\textrm {CTT}$ . We begin our case by arguing against Linnebo and Rayo’s claim that $\textrm {CTT}$ sheds new philosophical light on set theory. We then argue that, while $\textrm {CTT}$ ’s type-restrictions are unjustifiable, the type-restrictions imposed by ${\textrm {STT}}$ are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for $\textrm {CTT}$ . We end by examining an alternative approach to cumulative types due to Florio and Jones [10]; we argue that their theory is best seen as a misleadingly formulated version of ${\textrm {STT}}$ .
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